Split Courant algebroids as \(L_\infty\)-structures (Q778999)

A one-to-one correspondence between split Courant algebroids and multiplicative curved \(L_\infty\)-algebras is constructed and some of its properties are studied. In Section 2, the main notions concerning (pre-)Courant algebroids and Nijenhuis morphisms on (pre-)Courant algebroids are recalled. In Section 3, a review on \(L_\infty\)-algebras and Nijenhuis forms is given. In Section 4, the main result of the paper, namely, the construction of a map that establishes a one-to-one correspondence between split Courant algebroids and curved \(L_\infty\)-algebras, is presented. In Section 5, it is proved that this one-to-one correspondence preserves deformations by Nijenhuis operators. Moreover, some Nijenhuis morphisms on Courant algebroids are characterized as Nijenhuis forms on curved \(L_\infty\)-algebras. In Section 6, the twisting of a split Courant algebroid and of a curved \(L_\infty\)-algebra by a bivector are discussed. Furthermore, it is shown that the one-to-one correspondence preserves these twisting operations. In Section 7, the operations of twisting and deformation on both (pre-)Courant algebroids and curved (pre-)\(L_\infty\)-algebras are combined and some properties are pointed out.